1Complex numbers
IA Vectors and Matrices
1.4 Complex logarithm and power
Definition
(Complex logarithm)
.
The complex logarithm
w
=
log z
is a solution
to
e
ω
=
z
, i.e.
ω
=
log z
. Writing
z
=
re
iθ
, we have
log z
=
log
(
re
iθ
) =
log r
+
iθ
.
This can be multi-valued for different values of
θ
and, as above, we should select
the θ that satisfies −π < θ ≤ π.
Example. log 2i = log 2 + i
π
2
Definition
(Complex power)
.
The complex power
z
α
for
z, α ∈ C
is defined as
z
α
=
e
α log z
. This, again, can be multi-valued, as
z
α
=
e
α log |z|
e
iαθ
e
2inπα
(there
are finitely many values if
α ∈ Q
, infinitely many otherwise). Nevertheless, we
make z
α
single-valued by insisting −π < θ ≤ π.